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=> Algebraic Curves-Mordell Curve
=> Algebraic Curves-Ochoa Curve
=> Algebraic Integer
=> Algebraic Number
=> Algebraic Number Theory
=> Chebotarev Density Theorem
=> Class Field
=> Cyclotomic Field
=> Dedekind Ring
=> Fractional Ideal
=> Global Field
=> Local Field
=> Number Field Signature
=> Picard Group
=> Pisot Number
=> Weyl Sum
=> Casting Out Nines
=> A-Sequence
=> Anomalous Cancellation
=> Archimedes' Axiom
=> B2-Sequence
=> Calcus
=> Calkin-Wilf Tree
=> Egyptian Fraction
=> Egyptian Number
=> Erdős-Straus Conjecture
=> Erdős-Turán Conjecture
=> Eye of Horus Fraction
=> Farey Sequence
=> Ford Circle
=> Irreducible Fraction
=> Mediant
=> Minkowski's Question Mark Function
=> Pandigital Fraction
=> Reverse Polish Notation
=> Division by Zero
=> Infinite Product
=> Karatsuba Multiplication
=> Lattice Method
=> Pippenger Product
=> Reciprocal
=> Russian Multiplication
=> Solidus
=> Steffi Problem
=> Synthetic Division
=> Binary
=> Euler's Totient Rule
=> Goodstein Sequence
=> Hereditary Representation
=> Least Significant Bit
=> Midy's Theorem
=> Moser-de Bruijn Sequence
=> Negabinary
=> Negadecimal
=> Nialpdrome
=> Nonregular Number
=> Normal Number
=> One-Seventh Ellipse
=> Quaternary
=> Radix
=> Regular Number
=> Repeating Decimal
=> Saunders Graphic
=> Ternary
=> Unique Prime
=> Vigesimal
Ziyaretçi defteri
 

Goodstein Sequence

Given a hereditary representation of a number n in base b, let B[b](n) be the nonnegative integer which results if we syntactically replace each b by b+1 (i.e., B[b] is a base change operator that 'bumps the base' from b up to b+1). The hereditary representation of 266 in base 2 is

266 = 2^8+2^3+2
(1)
= 2^(2^(2+1))+2^(2+1)+2,
(2)

so bumping the base from 2 to 3 yields

 B[2](266)=3^(3^(3+1))+3^(3+1)+3.
(3)

Now repeatedly bump the base and subtract 1,

G_0(266) = 266
(4)
= 2^(2^(2+1))+2^(2+1)+2
(5)
G_1(266) = B[2](266)-1=3^(3^(3+1))+3^(3+1)+2
(6)
G_2(266) = B[3](G_1)-1=4^(4^(4+1))+4^(4+1)+1
(7)
G_3(266) = B[4](G_2)-1=5^(5^(5+1))+5^(5+1)
(8)
G_4(266) = B[5](G_3)-1=6^(6^(6+1))+6^(6+1)-1
(9)
= 6^(6^(6+1))+5·6^6+5·6^5+...+5·6+5
(10)
 
(11)
G_5(266) = B[6](G_4)-1
(12)
= 7^(7^(7+1))+5·7^7+5·7^5+...+5·7+4,
(13)
 
(14)

etc.

Starting this procedure at an integer n gives the Goodstein sequence {G_k(n)}. Amazingly, despite the apparent rapid increase in the terms of the sequence, Goodstein's theorem states that G_k(n) is 0 for any n and any sufficiently large k. Even more amazingly, Paris and Kirby showed in 1982 that Goodstein's theorem is not provable in ordinary Peano arithmetic (Borwein and Bailey 2003, p. 35).

 

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